Answer: vertical translation
Explanation:
A function is even if:
f(x) = f(-x)
and a function is odd if:
f(-x) = -f(x).
Now, we want to find a translation that only affects the symmetry of an odd function (but not that one of the even function).
The correct answer will be a vertical translation, now let's prove it.
when we have a function f(x) and we want to transform it into g(x), such that g(x) is a vertical translation of (for example) A units up, we can write the relation as:
g(x) = f(x) + A.
Now, let's see if this preserves the symmetry of the original function.
1) if f(x) is even, then:
g(x) = f(x) + A
g(-x) = f(-x) + A
But we know that f(-x) = f(x)
then:
g(-x) = f(-x) + A = f(x) + A = g(x)
this means that g(x) is even, the symmetry of an even function is not affected by this transformation.
Now suppose that f(x) is odd.
Then:
g(x) = f(x) + A
g(-x) = f(-x) + A
and f(-x) = -f(x)
then:
g(-x) = f(-x) + A = -f(x) + A
and this is clearly different than g(x) = f(x) + A.
Then the odd symmetry is broken.